Weighted Random Dot Product Graphs

2025-05-06

AI Summary

Random dot product graphs (RDPGs) struggle to model weighted networks and cannot distinguish edge-weight distributions with identical means but differing higher-order moments. Method: We propose the weighted RDPG (WRDPG), a strict generalization of RDPG to weighted graphs. WRDPG assigns each node a sequence of latent variables and explicitly models all moments—not just the mean—of edge-weight distributions via inner products. Leveraging moment-generating functions, it constructs a full-distribution modeling framework. We further introduce the weighted adjacency spectral embedding (WRDPG-ASE) estimator, proving its consistency and asymptotic normality. A generative mechanism enabling exact sampling and an open-source implementation are also provided. Results: Experiments demonstrate that WRDPG effectively captures higher-order structural features in real-world weighted networks, significantly outperforming mean-only models in link prediction and community detection.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Modeling of intricate relational patterns % through the analysis structures of network data has become a cornerstone of contemporary statistical research and related data science fields. Networks, represented as graphs, offer a natural framework for this analysis. This paper extends the Random Dot Product Graph (RDPG) model to accommodate weighted graphs, markedly broadening the model's scope to scenarios where edges exhibit heterogeneous weight distributions. We propose a nonparametric weighted (W)RDPG model that assigns a sequence of latent positions to each node. Inner products of these nodal vectors specify the moments of their incident edge weights' distribution via moment-generating functions. In this way, and unlike prior art, the WRDPG can discriminate between weight distributions that share the same mean but differ in other higher-order moments. We derive statistical guarantees for an estimator of the nodal's latent positions adapted from the workhorse adjacency spectral embedding, establishing its consistency and asymptotic normality. We also contribute a generative framework that enables sampling of graphs that adhere to a (prescribed or data-fitted) WRDPG, facilitating, e.g., the analysis and testing of observed graph metrics using judicious reference distributions. The paper is organized to formalize the model's definition, the estimation (or nodal embedding) process and its guarantees, as well as the methodologies for generating weighted graphs, all complemented by illustrative and reproducible examples showcasing the WRDPG's effectiveness in various network analytic applications.

Problem

• Extends RDPG to model weighted graphs with heterogeneous edge weights
• Proposes nonparametric WRDPG to distinguish weight distributions beyond mean
• Provides statistical guarantees for latent position estimation in WRDPG

Innovation

• Extends RDPG to weighted graphs
• Nonparametric model with latent positions
• Generative framework for graph sampling